### Degenerating Einstein spaces

- Series
- PDE Seminar
- Time
- Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Ruobing Zhang – Stony Brook University – ruobing.zhang@stonybrook.edu

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- Series
- PDE Seminar
- Time
- Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Ruobing Zhang – Stony Brook University – ruobing.zhang@stonybrook.edu

In the talk we discuss singularity formation of Einstein metrics
as the underlying spaces degenerate or collapse. The usual analytic tools
such as uniform Sobolev inequalities and nonlinear a priori estimates are
unavailable in this context. We will describe an entirely new way to
handle these difficulties, and construct degenerating Ricci-flat metrics
with quantitative singularity behaviors.

- Series
- PDE Seminar
- Time
- Tuesday, October 22, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Philippe G. LeFloch – Sorbonne University and CNRS – contact@philippelefloch.org

I will present a new method of analysis for Einstein’s

constraint equations, referred to as the Seed-to-Solution Method, which

leads to the existence of asymptotically Euclidean manifolds with

prescribed asymptotic behavior. This method generates a (Riemannian)

Einstein manifold from any seed data set consisting of (1): a Riemannian

metric and a symmetric two-tensor prescribed on a topological manifold

with finitely many asymptotically Euclidean ends, and (2): a density

field and a momentum vector field representing the matter content. By

distinguishing between several classes of seed data referred to as tame

or strongly tame, the method encompasses metrics with the weakest

possible decay (infinite ADM mass) or the strongest possible decay

(Schwarzschild behavior). This analysis is based on a linearization of

the Einstein equations (involving several curvature operators from

Riemannian geometry) around a tame seed data set. It is motivated by

Carlotto and Schoen’s pioneering work on the so-called localization

problem for the Einstein equations. Dealing with manifolds with possibly

very low decay and establishing estimates beyond the critical level of

decay requires significantly new ideas to be presented in this talk. As

an application of our method, we introduce and solve a new problem,

referred to as the asymptotic localization problem, at the critical

level of decay. Collaboration with T. Nguyen. Blog: philippelefloch.org

- Series
- PDE Seminar
- Time
- Tuesday, October 1, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Tongseok Lim – ShanghaiTech University – Tlim@shanghaitech.edu.cn

We study the geometry of minimizers of the interaction energy functional. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to be resolved by the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique up to rotation and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.

- Series
- PDE Seminar
- Time
- Tuesday, September 24, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Maja Taskovic – Emory University – maja.taskovic@emory.edu

In kinetic theory, a large system of particles is described by the particle density function. The Landau equation, derived by Landau in 1936, is one such example. It models a dilute hot plasma with fast moving particles that interact via Coulomb interactions. This model does not include the effects of Einstein’s theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. These effects are captured by the relativistic Landau equation, which was derived by Budker and Beliaev in 1956.

We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel in the relativistic collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain.

- Series
- PDE Seminar
- Time
- Tuesday, September 17, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Cheng Yu – University of Florida – chengyu@ufl.edu

In this talk, I will discuss from a mathematical viewpoint some sufficient conditions that guarantee the energy equality for weak solutions. I will mainly focus on a fluid equation example, namely the inhomogeneous Euler equations. The main tools are the commutator Lemmas. This is a joint work with Ming Chen.

- Series
- PDE Seminar
- Time
- Friday, July 26, 2019 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Jiayu Li – University of Science and Technology of China – jiayuli@ustc.edu.cn

In this talk we will review compactness results and singularity theorems related to harmonic maps. We first talk about maps from Riemann surfaces with tension fields bounded in a local Hardy space, then talk about stationary harmonic maps from higher dimensional manifolds, finally talk about heat flow of harmonic maps.

- Series
- PDE Seminar
- Time
- Tuesday, April 30, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- skiles 006
- Speaker
- Chenchen Mou – UCLA – muchenchen@math.ucla.edu

In this talk we study master equations arising from mean field game

problems, under the crucial monotonicity conditions.

Classical solutions of such equations require very strong technical

conditions. Moreover, unlike the master equations arising from mean

field control problems, the mean field game master equations are

non-local and even classical solutions typically do not satisfy the

comparison principle, so the standard viscosity solution approach seems

infeasible. We shall propose a notion of weak solution for such

equations and establish its wellposedness. Our approach relies on a new

smooth mollifier for functions of measures, which unfortunately does not

keep the monotonicity property, and the stability result of master

equations. The talk is based on a joint work with Jianfeng Zhang.

- Series
- PDE Seminar
- Time
- Tuesday, April 16, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Li Chen – University of Mannheim – pdebird@gmail.com

I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.

- Series
- PDE Seminar
- Time
- Tuesday, April 9, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Professor Gieri Simonett – Vanderbilt University – gieri.simonett@vanderbilt.edu

I will consider the motion of a rigid body with an interior cavity that is completely filled with a viscous fluid. The equilibria of the system will be characterized and their stability properties are analyzed. It will be shown that the fluid exerts a stabilizing effect, driving the system towards a state where it is moving as a rigid body with constant angular velocity. In addition, I will characterize the critical spaces for the governing evolution equation, and I will show how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity and stability properties for the system. The approach is based on the theory of Lp-Lq maximal regularity. (Joint work with G. Mazzone and J. Prüss).

- Series
- PDE Seminar
- Time
- Tuesday, April 2, 2019 - 15:00 for 1 hour (actually 50 minutes)
- Location
- skiles 006
- Speaker
- Professor Yan Guo – Brown University – Yan_Guo@Brown.edu

In a joint work with Sameer Iyer, the validity of steady Prandtl layer expansion is established in a channel. Our result covers the celebrated Blasius boundary layer profile, which is based on uniform quotient estimates for the derivative Navier-Stokes equations, as well as a positivity estimate at the flow entrance.

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